IZhO 2018 Problem 6

In a circle with a radius $R$ a convex hexagon is inscribed. The diagonals $AD$ and $BE$ , $BE$ and $CF$ , $CF$ and $AD$ of the hexagon intersect at the points $M$ , $N$ and $K$ , respectively. Let $r_1,r_2,r_3,r_4,r_5,r_6$ be the radii of circles inscribed in triangles $ABM,BCN,CDK,DEM,EFN,AFK$ respectively. Prove that. $$r_1+r_2+r_3+r_4+r_5+r_6\leq R\sqrt{3}$$ .